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Triangle wave
A triangle wave is a non-sinusoidal waveform named for its triangular shape. It is a periodic, piecewise linear, continuous real function. Like a square wave, the triangle wave contains only odd harmonics. However, the higher harmonics roll off much faster than in a square wave (proportional to the inverse square of the harmonic number as opposed to just the inverse). Harmonics for a mathematical description. ]] It is possible to approximate a triangle wave with additive synthesis by summing odd harmonics of the fundamental while multiplying every other odd harmonic by −1 (or, equivalently, changing its phase by π) and multiplying the amplitude of the harmonics by one over the square of their mode number, , (which is equivalent to one over the square of their relative frequency to the fundamental). The above can be summarised mathematically as follows: : \begin{align} x_\mathrm{triangle}(t) & {} = \frac\pi4\sum_{i=0}^{N - 1} (-1)^i n^{-2} \left(\sinf_0 n t \right) \end{align} where is the number of harmonics to include in the approximation, is the independent variable (e.g. time for sound waves), f_0 is the fundamental frequency, and is the harmonic label which is related to its mode number by n = 2i + 1 . This infinite Fourier series converges to the triangle wave as tends to infinity, as shown in the animation. Definitions , square, triangle, and sawtooth waveforms]] Another definition of the triangle wave, with range from -1 to 1 and period p, is: : x(t)=\frac{4}{p} \left (t-\frac{p}{2} \left \lfloor\frac{2 t}{p}+\frac{1}{2} \right \rfloor \right )(-1)^\left \lfloor\frac{2 t}{p}+\frac{1}{2} \right \rfloor where the symbol \lfloor n \rfloor is the floor function of n''. Also, the triangle wave can be the absolute value of the sawtooth wave: : x(t)= 2\left | {t \over p} - \left \lfloor {t \over p} + {1 \over 2} \right \rfloor \right | or, for a range from -1 to +1: : x(t)= 2 \left | 2 \left ( {t \over p} - \left \lfloor {t \over p} + {1 \over 2} \right \rfloor \right) \right | - 1. The triangle wave can also be expressed as the integral of the square wave: : x(t) = \int_0^t \sgn(\sin(u))\,du. Here is a simple equation with a period of 4 and initial value y(0) = 1 : : y(x) = |x\,\bmod\,4 - 2|-1. As this only uses the modulo operation and absolute value, this can be used to simply implement a triangle wave on hardware electronics with less CPU power. The previous equation can be generalized for a period of p, amplitude a, and initial value y(0) = a/2 : : y(x) = \frac{2a}{p} \biggl | \left( x \bmod p \right) - \frac{p}{2}\biggr | - \frac{2a}{4}. The former function is a specialization of the latter for a=2 and p=4: : y(x) = \frac{2 \times 2}{4} \biggl | \left( x \bmod 4 \right) - \frac{4}{2}\biggr | - \frac{2 \times 2}{4} \Leftrightarrow : y(x) = | \left( x \bmod 4 \right) - 2| - 1. An odd version of the first function can be made, just shifting by one the input value, which will change the phase of the original function: : y(x) = |(x-1)\,\bmod\,4 - 2|-1. Generalizing this to make the function odd for any period and amplitude gives: : y(x) = \frac{4a}{p} \biggl | \left( (x - \frac{p}{4}) \bmod p \right) - \frac{p}{2}\biggr | - a. In terms of sine and arcsine with period ''p and amplitude a: : y(x) = \frac{2a}{\pi}\arcsin\left(\sin\left(\frac{2\pi}{p}x\right)\right). Arc length The arc length per period "s" for a triangle wave, given the amplitude "a" and period length "p": : s = \sqrt{(4a)^2 + p^2}. See also * List of periodic functions * Sawtooth wave * Sound * Triangle function * Wave * Zigzag References * Category:Fourier series Category:Waveforms